Multi-Dimensional Scheduling for Real-Time Tasks on Heterogeneous Clusters

Transcription

1 Zhu XM, Lu PZ. Multi-dimensional scheduling for real-time tasks on heterogeneous clusters. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 24(3): May 2009 Multi-Dimensional Scheduling for Real-Time Tasks on Heterogeneous Clusters Xiao-Min Zhu ( ) and Pei-Zhong Lu ( ), Member, IEEE School of Computer Science, Fudan University, Shanghai , China {xmzhu, pzlu}@fudan.edu.cn Received October 3, 2008; revised January 25, Abstract Multiple performance requirements need to be guaranteed in some real-time applications such as multimedia data processing and real-time signal processing in addition to timing constraints. Unfortunately, most conventional scheduling algorithms only take one or two dimensions of them into account. Motivated by this fact, this paper investigates the problem of providing multiple performance guarantees including timeliness, QoS, throughput, QoS fairness and load balancing for a set of independent tasks by dynamic scheduling. We build a scheduler model that can be used for multi-dimensional scheduling. Based on the scheduler model, we propose a heuristic multi-dimensional scheduling strategy, MDSS, consisting of three steps. The first step can be of any existing real-time scheduling algorithm that determines to accept or reject a task. In step 2, we put forward a novel algorithm MQFQ to enhance the QoS levels of accepted tasks, and to make these tasks have fair QoS levels at the same time. Another new algorithm ITLB is proposed and used in step 3. The ITLB algorithm is capable of balancing load and improving throughput of the system. To evaluate the performance of MDSS, we perform extensive simulation experiments to compare MDSS strategy with MDSR strategy, DASAP and DALAP algorithms. Experimental results show that MDSS significantly outperforms MDSR, DASAP and DALAP. Keywords clusters, scheduling, multi-dimensional, heterogeneous, real-time, makespan 1 Introduction Clusters, consisting of multiple nodes interconnected by high-speed networks, have developed into efficient platforms characterized by cost-effectiveness, specializing in processing computing-intensive and dataintensive applications [1]. Specially, heterogeneous clusters receive a good deal of attention in practice. That is because the nodes with different powers purchased in different periods of time are usually combined in a cluster for economical purpose [2]. Meanwhile, a growing number of real-time applications have been developed and deployed in heterogeneous clusters, in which the correctness of the real-time applications depends not only on the logic results of computation, but also on the time instants at which these results are produced [3]. Examples of real-time applications include aircraft control, signal processing, image processing, etc. Those real-time systems, in which missing a deadline may be catastrophic, such as automated flight control systems are called hard real-time systems. There is another category called soft real-time systems, which are systems such as multimedia, where nothing catastrophic happens if some deadlines are missed [4]. In this paper, the system we consider belongs to the latter sort. In addition to timing requirements, many real-time applications have other important demands such as QoS, load balancing, QoS fairness and throughput. For example, in cluster-based real-time signal processing systems, signal processing has some critical characteristics [5,6]. 1) The signal data is able to be divided into independent task blocks to be parallel processed in different nodes. 2) The signal data must be processed in a certain timing scope. 3) Several algorithms can be used to process the signal data, as for decoding of block turbo codes, algorithms such as proposed in the literature [8-10] can all be employed. It is noted that high-complexity algorithms generally can guarantee signal processing at higher QoS level at the expense of long processing time, but low-complexity algorithms are just the opposite. 4) The conterminous two tasks should be concatenated one by one after Regular Paper This work is supported by the National Natural Science Foundation of China under Grant No , and the Special Funds of Authors of Excellent Doctoral Dissertation in China under Grant No The previous version appeared in Proceedings of the 10th IEEE International Conference on High Performance Computing and Communications (HPCC 2008), pp

2 Xiao-Min Zhu et al.: Multi-Dimensional Scheduling for Real-Time Tasks on Heterogeneous Clusters 435 processing. So, minimizing the finish time of the last executed task can improve throughput and decrease delay. 5) Fair QoS are required to make the processing quality smoother. 6) To ensure fast processing and good utilization of the system, load balancing is necessarily considered. Other real-time applications such as real-time image processing also have the similar characteristics. Consequently, it is significant to provide multi-dimensional performance guarantees for these practical applications [7]. Growing evidence shows that scheduling is an efficient method of achieving high performance for applications running on clusters [2]. However, to the best of our knowledge, most conventional scheduling algorithms only consider one or two dimensions among needed metrics, which presents the challenge of devising novel scheduling algorithms to supply the gap. Motivated by the above argument, we investigate in this paper a Multi-Dimensional Scheduling Strategy, MDSS, considering timing constraints, QoS, throughput, load balancing and QoS fairness on heterogeneous clusters. In this paper, QoS refers to tasks quality after being proceeded. The MDSS strategy uses three steps to realize our scheduling objectives. Step 1 employs the existing DASAP algorithm reported in [2] to decide the allocation of real-time tasks on the whole. Note that other real-time algorithms can also be used in the first step. In step 2, we propose a novel QoS optimization algorithm by optimizing QoS benefit, i.e., making tasks have Maximal QoS and Fair QoS (MQFQ for short). Step 2 adjusts accepted tasks allocation on the basis of step 1 and guarantees that the timing constraints of these tasks still can be satisfied. Last, step 3 employs a new proposed algorithm that minimizes the standard deviation of nodes latest finish time to Improve Throughput and Load Balancing (ITLB for short). The third step further tunes the allocation of accepted tasks without violating their timing constraints and changing their QoS levels. A key issue in MDSS strategy is that the latter steps are carried out based on the former steps and cannot degrade the performance determined by the former ones. Therefore, the three steps are closely interacted, not simply stacked together. The remainder of the paper is organized as follows. Related work in this area is reviewed in Section 2. In Section 3, we present a real-time scheduler model and a task model with multi-dimensional performance requirements. Section 4 proposes a scheduling strategy, MDSS, including two novel algorithms MQFQ and ITLB, and investigates their properties. Performance evaluation is presented and discussed in Section 5. Finally, Section 6 concludes the paper with a summary and future work. 2 Related Work Study on scheduling problem in cluster computing has received a good deal of attention. Unfortunately, many practical instances of scheduling algorithms have been found to be NP-complete [11], i.e., it is believed that there is no optimal polynomial-time algorithm for them. These negative results motivated the need for heuristic approaches to develop scheduling algorithms. Up to now, many heuristic scheduling algorithms have been proposed, e.g., Braun et al. evaluated eleven commonly used algorithms OLB, UDA, Fast Greedy, Min-min, Max-min, Greedy, Genetic Algorithm (GA), Simulated Annealing (SA), GSA, Tabu and A [12]. The experimental results form literature [12] show that Minmin, GA and A* are able to deliver good performance. Maheswaran et al. proposed a Sufferage heuristic that has better performance than Min-min [13], etc. Although these algorithms can achieve high performance for non-real-time applications, they are not fit for realtime applications because of the lack of guarantee for real-time tasks to meet their deadlines. In particular, the artificial intelligence approaches exhibit highly variable computation time, hence, adopting their worst case execution time perhaps results in an unacceptable under-utilization or in a non-schedulable system [14]. The problem of real-time scheduling has been extensively studied based on heuristic in recent years. Real-time scheduling algorithms can be either static (off-line) or dynamic (on-line). In static algorithms, the assignment of tasks to processors (nodes) and the time at which the tasks start execution are determined a priori [15]. Static algorithms are often used to schedule periodic tasks like those presented in literature [16, 17]. However, those aperiodic tasks whose characteristics are not known a priori must employ dynamic scheduling algorithms, e.g., see [2, 3, 5 7, 18, 19]. Our work is focused on dynamically scheduling aperiodic tasks. In addition, several algorithms schedule tasks with precedence constraints, which are often represented by directed acyclic graphs (DAG) [2,16,20]. In contrast, we consider in this paper tasks having no precedence constraints because tasks with precedence constraints can be scheduled by transforming the precedence graph into independent nodes with new ready time and deadlines [21]. Moreover, some scheduling algorithms of real-time tasks are preemptive [22,23], whereas our scheduling algorithm is non-preemptive like that reported in [24], i.e., when a task starts to execute on a processor, no other tasks interrupt its execution, which is more efficient, particularly suitable for soft real-time

3 436 J. Comput. Sci. & Technol., May 2009, Vol.24, No.3 applications than the preemptive approaches due to reducing the overheads needed for switching among tasks [25]. Recently, some effort has been devoted to develop dynamic and non-preemptive scheduling algorithms for real-time, independent and aperiodic tasks, e.g., Manimaran et al. proposed a dynamic scheduling algorithm which exploits parallelism in tasks in order to meet their deadlines, thereby increasing the performance of the system [26]. Xie and Qin studied a scheduling algorithm considering security requirements (a kind of QoS requirement) of real-time applications to enhance the security of systems [3]. Abdelzaher et al. presented real-time middleware services for a flight control application by QoS negotiation [27]. However, these scheduling algorithms are designed for homogeneous systems, making them unsuitable for heterogeneous computing environment. With the wide use of heterogeneous clusters, some work has been done to combine real-time computing with heterogeneous systems, e.g., Guo et al. studied two scheduling algorithms by predicting CPU load for each multimedia task so as to achieve load balancing in a heterogeneous cluster-based web server [28]. Harada et al. proposed an adaptive resource allocation method on the basis of conventional control theory to provide fair QoS level for real-time independent tasks [29]. However, all the algorithms or strategies mentioned above only consider part of performance metrics in terms of timing constraints, QoS, throughput, load balancing and QoS fairness. Ignoring some critical factors may degrade any additional performance gained by using cluster computing. He et al. addressed a dynamic scheduling of parallel jobs with QoS demands and proposed a new scheduler TITAN that aims at improving performance measured by three metrics including over-deadline, makespan and idle-time [30]. Although the TITAN considers some metrics, it is used in homogeneous multiclusters and Grid environment, whereas our scheduling strategy focuses on heterogeneous clusters. Besides, TITAN does not take QoS fairness, throughput and load balancing into account. Doǧan and Özgüner investigated the problem of scheduling a set of independent tasks with multiple QoS needs, which include timeliness, reliability, security, data accuracy, and priority, in a heterogeneous computing system [31]. However, the metrics considered are different from ours. In addition, the proposed scheduling algorithm QSMTS IP is static but ours is dynamic. In this paper, we concentrate on a novel dynamic scheduling strategy that guarantees multiple performance needs for soft real-time tasks. One of our scheduling objectives is to accept real-time tasks as many as possible so as to guarantee high schedulability. Besides, we strive to enhance the QoS levels of accepted tasks and make them have fair QoS levels at the same time. It is noted that the minimum QoS level is acceptable for each real-time tasks. The last objective is to balance the system load and achieve high throughput. 3 Scheduler Model and Task Model 3.1 Scheduler Model Generally, the scheduler model can be either distributed or centralized. In a distributed scheduler model, tasks arrive independently at each local scheduler, which produces schedules in parallel with other schedulers. In a centralized scheduler model, all tasks arrive at a central processor called scheduler, from which they are distributed to nodes in a cluster for further execution. In this paper, we adopt the centralized sort. The reason is that the centralized scheduler model has two attractive features compared with the distributed scheme [2]. First, it directly provides the centralized scheduler with fault-tolerance using a backup scheduler. Second, implementation of a centralized scheduler model is simpler and easier than that of distributed scheduler model. In order to realize our scheduling objectives, we modify the traditional centralized scheduler model and propose a novel scheduler model illustrated in Fig.1. Fig.1. Scheduler model. The scheduler is modeled by a real-time controller, a QoS level controller and a balancing controller. The scheduler and nodes are connected by a high-speed network and each node maintains a local queue in which accepted tasks are queuing up for execution on the nodes. In this paper, we build the real-time controller using DASAP algorithm [2]. Other real-time scheduling algorithms can also be employed, as well. The real-time controller is used to accept or reject a real-time task waiting in the scheduler queue on condition that if the deadline and minimum QoS level of the task can be guaranteed. If so, the task will be transferred into the accepted queue, otherwise, it will be dropped into the

4 Xiao-Min Zhu et al.: Multi-Dimensional Scheduling for Real-Time Tasks on Heterogeneous Clusters 437 rejected queue. The QoS controller aims at maximizing the QoS levels of accepted tasks, and making these tasks have fair QoS levels at the same time. Following the QoS controller, the balancing controller will further adjust these tasks allocation for load balancing and high throughput without changing their QoS levels and missing their deadlines. Finally, these tasks are dispatched to designated nodes for execution. 3.2 Task Model In this paper, we consider a set T = {t 1, t 2,..., t n } of independent real-time tasks. There is no communication among tasks while executing, and one task can only execute on one node. A heterogeneous cluster is composed of a set N = {n 1, n 2,..., n m } of heterogeneous nodes with different processing powers connected by high-speed interconnection networks such as Myrinet and InfiniBand. The execution time matrix is denoted as E = {e ij } n m, where element e ij is the expected execution time of task t i on node n j. It is assumed that the expected execution time is known. Techniques such as code profiling/analytic benchmarking and statistical prediction have been devised to estimate e [12] ij. Let EST = (est ij ) n m, where element est ij denotes the earliest start time of task t i on node n j. f i is the finish time of task t i. Let Z = (z ij ) n m be a binary matrix, where element z ij equals 1 if and only if t i has been assigned to n j ; otherwise z ij = 0. CE = (ce ij ) n m is also a binary matrix, where element ce ij equals 1 if and only if the QoS level of t i can be enhanced on n j ; otherwise ce ij = 0. Let QoS levels be a set Q, Q = {q 1, q 2,..., q k }, where q 1 < q 2 < < q k. X i represents all possible schedules for task t i, i.e., there are several nodes that can be chosen as candidate nodes to which t i is assigned. x i X i is a scheduling decision of t i. The QoS level of task t i adopting schedule x i can be denoted by q(x i ). x i is a feasible schedule if 1) deadline of d i can be guaranteed, i.e., f i d i, and 2) the QoS requirement is satisfied, i.e., q 1 q(x i ) q k. In order to accept real-time tasks as many as possible, each task is assigned minimum QoS level at the beginning. Then, the QoS controller strives to enhance QoS levels of accepted tasks and makes them have fair QoS levels. The QoS controller adopts a policy to measure QoS benefits. The maximum QoS benefits of tasks on node n j should satisfy the following two objectives under timing constraints: { max α = x i X i ( n / n ) } (z ij q(x i )) z ij, (1) i=1 i=1 { min β = x i X i (( n (z ij q(x i ) α) 2)/ n ) } z ij. i=1 i=1 (2) The first goal is to maximize the QoS levels of tasks on the same nodes, and the second goal is to make the difference of QoS levels minimum, which leads to fair QoS levels for them. Definition 1. The QoS benefit of tasks on node n j is defined as: α χ = ε + β, (3) where ε is a small positive real number used to make the denominator not zero. α and β are defined in (1) and (2), respectively. From the above equation, we can observe that a big value of α and a small value of β are able to make the value of χ big, which satisfies our objectives. Meanwhile, to make numerator have the same magnitude with denominator, we use β. Definition 2. If the QoS level of a task t k assigned to node n j by real-time controller is enhanced, the new QoS benefit of tasks on node n j is defined as: χ k = α ε + β. (4) Our QoS level enhancement method is designed to increase tasks QoS benefits each time in the following way: max{χ k}( ce kj = 1 χ k χ) ( ce kj = 1 χ k < χ). (5) Then, the task selected to enhance its QoS level is the task t l such that: χ l = max{χ k}, z kj = z lj = 1. (6) For the convenience of reference in the rest of the paper, we summarize the notation of task model in Table 1. Parameters e ij est ij z ij ce ij x i q(x i ) χ χ k Table 1. Task Model Parameters Explanation Execution time of task t i on node n j The earliest start time of task t i on node n j z ij = 1 if and only if t i has been assigned to n j ; otherwise z ij = 0 ce ij = 1 if and only if the QoS level of t i can be enhanced on n j ; otherwise ce ij = 0 A feasible schedule, x i X i QoS level of task t i adopting schedule x i QoS benefit of tasks on node n j New QoS benefit when the QoS level of a task t k assigned to node n j is enhanced

5 438 J. Comput. Sci. & Technol., May 2009, Vol.24, No.3 Property 1. The QoS level enhancement method avoids the local extremum problem. In our study, we do not greedily make the new QoS benefit χ k larger than the former QoS benefit χ each time. That is because although tasks on the same node have fair QoS levels, sometimes, the QoS levels still can be enhanced. Now, we give two simple examples to explain the QoS level enhancement method. Example 1. Suppose there are five acceptable QoS levels (1, 2, 3, 4, 5) and five tasks (t 1, t 2, t 3, t 4, t 5 ) on a node. The corresponding QoS levels of the five tasks now are 2, 3, 5, 2 and 1, respectively. Let ε be 0.1. The value of χ is If the QoS level of each task is enhanced by 1 (except t 3, because t 3 already has the highest QoS level currently), the new values are χ 1 = 1.96, χ 2 = 1.78, χ 4 = 1.96, and χ 5 = max{χ 1, χ 2, χ 4, χ 5) = So, task t 5 is selected to enhance its QoS level. The new QoS levels of the five tasks are 2, 3, 5, 2 and 2, respectively. Example 2. Suppose there are the same task number and acceptable QoS levels as those in Example 1. ε is still 0.1. The difference is that now the QoS levels of the five tasks are all 1. We can calculate that χ = 10. If the QoS level of any task is enhanced by 1, the new QoS benefit is 2.4. Although χ k < χ (k = 1, 2, 3, 4, 5), we still need to improve the QoS levels if enhancing the QoS level of a task will not miss its deadline. Therefore, the new QoS level of one of the five tasks is 2 and others are 1. Obviously, Example 2 shows that our QoS level enhancement method efficiently avoids the local extremum problem, making tasks have chance to further enhance their QoS levels under the constraints that their timing constraints are satisfied. Load balancing is another issue considered in our scheduling strategy. The finish time of node n j can be denoted as: LF j = max {z ijf i }, (7) 1 i n where f i is the finish time of task t i. The goal of load balancing is to make nodes have similar finish time. Hence, the degree of load balancing is able to be evaluated by the standard deviation of nodes finish time. m LB = (LF j LF ) 2/ m, (8) j=1 where LF = m j=1 LF j/m, and m is the node number. To achieve load balancing, we should try to minimize the value of LB. An additional major goal in task scheduling is high throughput. Makespan is a commonly used metric to measure throughput of a system. Makespan can be expressed as the finish time of the last executed task [30]. So the makespan M can be denoted as: M = max 1 j m {LF j}. (9) Algorithm ITLB shown in Subsection 4.2 will employ an approach to minimizing LB and M to achieve load balancing and high throughput. It is noted that minimizing LB and M cannot validate the timing constraints of accepted tasks and degrading their QoS levels. 4 Multi-Dimensional Scheduling Strategy MDSS Based on the scheduler model, we propose a novel scheduling strategy MDSS with three steps, which takes multiple scheduling objectives into consideration. In the first step, any real-time scheduling algorithm can be used to determine if a task can be accepted or rejected. In our study, we employ the DASAP algorithm proposed in literature [2] in step 1. Note that, every task is given the minimum QoS level while scheduling in step 1 to guarantee high schedulability. In the following subsections, we will present the two novel algorithms MQFQ and ITLB used in steps 2 and 3, respectively. 4.1 MQFQ Algorithm In Subsection 3.2, we proposed the objective to maximize the QoS benefits. For facilitating the presentation of MQFQ algorithm, it is necessary to introduce some properties and theorems before we introduce it. Property 2. If the QoS level of task t i on node n j can be enhanced, namely, ce ij = 1, the following inequalities must be satisfied: est ij + e ij (q(x i )) d i, (10) t k, s kj > s ij : est kj + e kj (q(x k )) d k, (11) where e ij (q(x i )) is the execution time of task t i on node n j with the QoS level q(x i ). s ij represents the execution sequence of t i on n j. est kj = est kj + it ij, where it ij is the increased execution time of task t i on n j, and est kj is the new earliest start time of task t k when the QoS level of t i is enhanced. The earliest start time of t i on n j can be calculated as follows: est ij = r j + where r j is the ready time of node n j. s kj <s ij e kj (q(x k )), (12)

6 Xiao-Min Zhu et al.: Multi-Dimensional Scheduling for Real-Time Tasks on Heterogeneous Clusters 439 The condition in Property 2 indicates that the enhancement of one task s QoS level cannot violate any deadlines of tasks that have been admitted to the same node. Theorem 1. Once a task s QoS level cannot be enhanced, the QoS level of this task cannot be enhanced any more. Proof. 1) If q(x i ) = q k, the QoS level of task t i is already the maximum. Thus, the QoS level of t i does not need to be enhanced any more. 2) If q(x i ) < q k, task t i s QoS level not being enhanced has the following two conditions: or t k, s kj > s ij : est kj + e kj (q(x k )) > d k, est ij + e ij (q(x i )) > d i. Suppose t p, p i, whose QoS level can be enhanced, (i) If s pj < s ij, then we get the following equation: t q, s qj > s pj : est qj = est qj + it pj, which results in that all the tasks whose execution sequences are later than that of t i have later earliest start time than before. If the QoS level of t i is enhanced by one, Property 2 absolutely cannot be satisfied because its QoS level enhancement has violated Property 2 when tasks whose execution sequences are later than that of t i have earlier earliest start time. (ii) If s pj > s ij, then t q, s qj > s pj : est qj = est qj + it pj, which leads to that some of tasks whose execution sequences are later than that of t i have later earliest start time than before. Similarly, this condition will make task t i have no chance to enhance its QoS level. From Theorem 1, we can obtain that once a task s QoS level cannot be enhanced, the task does not require the QoS level enhancement operation any more. Therefore, the time complexity of MQFQ algorithm is reduced. The pseudocode of MQFQ is given in Fig.2. The goal of the MQFQ algorithm is to maximize QoS benefits of tasks while guaranteeing real-time requirements for the tasks running on the same node. To avoid the local extremum problem, we do not let the QoS benefit monotonously increase. Thus, the MQFQ algorithm is designed to select the task whose QoS level enhancement results in maximum new QoS benefit regardless of whether the new QoS benefit is less than the old one or not (see lines 19 22). If the QoS level of a task is q k, the task will be deleted from set S (see lines 7, 8), because q k is the highest QoS level, that cannot be enhanced any more. Before increasing the QoS level of task t i on node n j, the MQFQ algorithm attempts to verify if the timing constraints of t i and those tasks whose execution sequences are later than that of t i can be satisfied (see line 12). If not, that indicates the task s QoS level cannot be enhanced any more, so t i is deleted from S (see line 14). If line 18 can be executed, it shows at least one task s QoS level can be enhanced. 1. for each node n j in the cluster do 2. S ; / Initialization / 3. Put each task t i on node n j into set S; / This allocation of t i on n j is determined by Step 1 / 4. while S do 5. max benefit 0; flag FALSE; / Initialization / 6. for each task t i with QoS level q m in set S do 7. if q m == q k then / t i has the highest QoS level / 8. Remove t i from S; / Theorem 1 / 9. continue; / to deal with next task in S / 10. else increase the QoS level of t i by 1; 11. Calculate the increased time it ij ; 12. if est ij + e ij (q m+1 ) > d i or t k, s kj > s ij : est kj + e kj (q(x k )) + it ij > d k (Property 2) then 13. Degrade the QoS level of t i by 1; / Rollback to the previous QoS level / 14. Remove t i from S; / Theorem 1 / 15. continue; / to deal with next task in S / 16. end if 17. end if 18. flag TRUE; / QoS level can be enhanced / 19. Calculate the new QoS benefit χ i using (4); 20. if χ i > max benefit then 21. max benefit χ i ; / find the maximum new QoS benefit / 22. end if 23. end for 24. if flag = = true then 25. Record the task t i with max benefit, selected task t i ; 26. Update the execution time of task selected task; 27. for each task t k whose execution sequence is later than that of selected task do 28. Update the est kj of task t k on node n j ; 29. end for 30. end if 31. end while 32. end for Fig.2. Pseudocode of MQFQ. Now, we evaluate the time complexity of MQFQ as follows. Theorem 2. The average time complexity of MQFQ is O(n 2 k/m), where n is the number of tasks, m is the number of nodes, and k is the number of QoS levels. Proof. The average time complexity of putting tasks assigned on the same node into set S is O(n/m) (line 3). To verify if a task s QoS level can be enhanced, the average time complexity is O(n/m) (line 12). The average time complexity of calculating the new QoS benefit χ i is O(n/m) (line 19). To update the earliest start time of tasks whose execution sequences are later than that of t i, the average time complexity is also O(n/m).

7 440 J. Comput. Sci. & Technol., May 2009, Vol.24, No.3 For other lines, they only consume O(1). Thus, the average time complexity of MQFQ is as follows: O(m)(O(n/m) + O(n/m)((O(k)(O(n/m))) = O(n 2 k/m). 4.2 ITLB Algorithm The ITLB algorithm is used to balance the workload and produce high throughput for the heterogeneous system. The core idea of ITLB algorithm is to move the tasks on the node with latest finish time to the node with earliest finish time. Property 3. A task being moved from a node to another node must satisfy three conditions: 1) the value of makespan is decreased; 2) the QoS level of this task cannot be changed; 3) the timing constraint of the task is guaranteed. Theorem 3. The system is regarded as load balancing and has best throughput if and only if no task can be moved according to Property 3. Proof. Suppose node n j has the latest finish time LF j, the last executed task on it is t i, and n k (k j) has the smallest LF k if t i has been moved to it. 1) If LF k + e ik (q(x i )) LF j, then n p, p k j: LF p +e ip (q(x i )) LF j, which results in the makespan, keeps invariable or increases. So the task t i cannot be moved. 2) If LF k + e ik (q(x i )) < LF j, but est ik + e ik (q(x)) > d i. Although the makespan is decreased, task t i cannot meet its deadline with previous QoS level. Thus the task t i cannot be moved. The pseudocode of ITLB is shown in Fig do 2. balanceflag TRUE; 3. Find the node n j, where LF j is latest; min LF LF j ; 4. Find the last executed task t i on node n j ; 5. for each node n k 6. if LF k + e ik (q(x i )) < min LF &&LF k + e ik (q(x i )) < d i (Property 3) then 7. min LF LF k ; 8. balanceflag FALSE; 9. end if 10. end for 11. if balanceflag==false then 12. Allocate task t i to node n q whose LF q = min LF ; 13. Update the values of LF q and LF j, LF q LF q +e iq (q(x i )), LF j LF j e ij (q(x i )); 14. end if 15. while balanceflag == FALSE Fig.3. Pseudocode of ITLB. To make the system load balancing and achieve high throughput, the ITLB algorithm attempts to move the last executed task on the node with latest finish time to the destination node with earliest finish time under the constraints of Property 3 (see lines 3 7). This method is able to decrease the value of makespan and decrease the difference of finish time among different nodes, i.e., improve the throughput and load balancing. The time complexity of ITLB can be evaluated as below. Theorem 4. The time complexity of ITLB is O(mn + n 2 /m), where m is the number of nodes, and n is the number of tasks. Proof. The time complexity of finding the node n j whose finish time is latest is O(m) (line 3). To find the last executed task on the selected node n j is O(n/m). Finding the node n q whose finish time is earliest is O(m) (lines 5 10). The time complexity of allocating task to node n q, and updating the finish time of n q and n j is O(1) (lines 12 and 13). Thus, the time complexity of ITLB can be calculated as follows: O(n)(O(m)+O(n/m)+O(m)+O(1)) = O(mn+n 2 /m). In the worst case, the time of do-while loop is n 1. Since the first step has already allocated tasks on node on the whole, the time of loop is significantly decreased. Consequently, the time complexity of ITLB is low. 5 Performance Evaluation In this section, we present several groups of experimental results obtained from extensive simulations to evaluate performance of the proposed scheduling strategy MDSS. A competitive advantage of conducting simulation experiments is that performance evaluation on a large-scale cluster can be accomplished without additional hardware cost [3]. To reveal performance improvements gained by our proposed strategy, we compare it with DASAP, DALAP algorithms proposed in the literature [2] and a strategy called MDSR which includes three steps in which steps 1 and 3 are the same as MDSS, but step 2 uses Round-Robin method to enhance QoS levels of tasks on the same node. To make the comparisons fair, we slightly modify the DASAP and DALAP algorithms in such a way that they arbitrarily pick a QoS level within the QoS range of tasks. Though both of the algorithms are intended to schedule real-time tasks with QoS requirements, they make no effort to maximize the QoS benefit, achieve load balancing and obtain high throughput. The DASAP and DALAP algorithms are briefly described as below. 1) DASAP (Dynamic Schedule As Soon As Possible): The task with earliest deadline is always executed first, and the task is allocated to the node on which it has the earliest start time.

8 Xiao-Min Zhu et al.: Multi-Dimensional Scheduling for Real-Time Tasks on Heterogeneous Clusters 441 2) DALAP (Dynamic Schedule As Late As Possible): The task with earliest deadline is always executed first, and the task is allocated to the node on which it has the latest start time subject to the constraints that deadlines of all tasks are guaranteed. In our experiments, to evaluate the performance of MDSS, we compare MDSS, MDSR, DASAP and DALAP in the following metrics: 1) Guarantee Ratio (GR) defined as: GR=Total number of tasks guaranteed to meet their deadlines/ Total number of tasks 100%; 2) QoS Benefit Average (QBA) which represents the average of QoS benefits on all nodes; 3) QoS Level Average (QLA) used to test the QoS levels of accepted tasks; 4) QoS Level Standard Deviation (QLSD) employed to measure the QoS fairness; 5) Makespan (MS) used to measure throughput; 6) Finish Time Standard Deviation (FTSD) of nodes to evaluate the degree of load balancing. 5.1 Simulation Method and Parameters Heterogeneity can be divided into node heterogeneity and task heterogeneity [12]. The node heterogeneity refers to the difference of a task s execution time on different nodes and the task heterogeneity is the difference of different tasks execution time on a node. Before presenting empirical results in detail, we present the simulation methods as follows. 1) In order to present the node heterogeneity, we use p j to represent the processing power of node n j. p j is a positive real number and the bigger the value of p j is, the better the processing power of n j is. Parameters poweraverage and powerspan denote the average processing power of all nodes and variable scope taking poweraverage as center, respectively. p j is uniformly distributed between poweraverage powerspan and poweraverage + powerspan. 2) We employ h i, a positive real number, to denote the hardness of task t i. The bigger the value of h i is, the longer the execution time of t i is. The execution time of tasks with bigger hardness is longer than that of tasks with smaller hardness on the same nodes. Parameters hardnessaverate and hardnessspan denote the average hardness of all tasks and variable scope taking hardnessaverage as center, respectively. Similarly, the value h i is uniformly distributed between hardnessaverage hardnessspan and hardnessaverage + hardnessspan. 3) Because our scheduling model is a general one. So, without loss of generality, we only need to give an abstract QoS level definition. As a result, we suppose that 0 q(x i ) 9 is the current QoS level of t i, and q(x i ) is a positive integer. 4) According to [12], the execution time matrix E = (e ij ) n m can be classified into two classes: consistent and inconsistent. For a consistent matrix E, if node n x has a less execution time than node n y for task t k, then the same is true for any task t i. For a inconsistent matrix E, if processor n x has a less execution time than processor n y for task t k, then the same is not true for other task t i. In our study, the matrix E belongs to the consistent type. The execution time e ij of task t i on node n j can be generated as: e ij = (1 + q(x i )/10) basetime (h i /p j ). Parameter basetime is a random positive real number. 5) The deadline d i of task t i is chosen as: d i = a i + max{e ij } + basedeadline, where a i is the arrival time, max{e ij } is the maximum execution time of t i on all nodes, and basedeadline is a random positive real number. basedeadline determines that tasks have loose deadlines or not. 6) The ready time t j of node n j is a random number, which is between 0 and readytime. Parameter ready- Time is a random positive real number. Table 2 summarizes the key parameters of the system in our experiments. Parameter Table 2. Parameters for Simulation Studies Node Number (27)-(15, 45, 5) Task Number (2000) poweraverage (700) Value(Fixed)-(min, max, step) powerspan (400)-(250, 550, 50) hardnessaverage (190) hardnessspan (100) basedeadline (50)-(20, 90, 10) basetime (3)-(1, 6, 0.5) readytime (9) ε (0.1) 5.2 Performance Impact of Node Number In this subsection, we present this group of experimental results to observe the performance comparison of MDSS, MDSR, DASAP and DALAP with respect to the impact of node number. Fig.4 illustrates the performance impact of node number. Fig.4(a) shows that MDSS and MDSR have the identical guarantee ratio (GR) and improve it over DASAP and DALAP by up to 16.79% and 13.54% in average, respectively. This result can be attributed to the fact that MDSS and MDSR both use DASAP algorithm and set the QoS levels of all tasks to the lowest (zero in our experiments) in step 1. However, DASAP and DALAP

9 442 J. Comput. Sci. & Technol., May 2009, Vol.24, No.3 Fig.4. Performance impact of node number. randomly pick up QoS levels for tasks. Therefore, MDSS and MDSR outperform DASAP and DALAP in terms of schedulability. In addition, MDSS and MDSR do not add and drop any task accepted by the first step in steps 2 and 3, so they are always have the same guarantee ratio. It is observed from Fig.4(a) that the GRs of all policies improve with the increasing value of the node number. That is because increasing the number of nodes enhances the computational capability of the system, which may in turn guarantee more tasks to be finished before their deadlines. Figs. 4(b) 4(d) show that QBA, QLA and DLSD of DASAP and DALAP approach constant values, whereas the same metrics of MDSS and MDSR are inclined to our objectives, i.e., high QoS levels average and low difference of QoS levels. These results are obvious because DASAP and DALAP do not use our QoS enhancement mechanism. Fig.4(b) plots that the QBA of MDSS maintains a small constant value when the node number is less than 25, which can be easily explained that if there are less nodes in the system resulting in heavy system load, MDSS regards the high schedulability as its main scheduling objective. So from Figs. 4(c) and 4(d), we can observe that MDSS has lower value of QLA and higher value of QLSD when the node number is less than 25. However, when the node number is more than 25, as for MDSS, the values of QBA, QLA extremely increase and QLSD decreases correspondingly. That is because MDSS is capable of employing slack time to improve the QoS levels of accepted tasks. When the node number increases more than 25, all tasks can be accepted and the timing constraints are much looser, so the QoS levels of all tasks can achieve the highest value, the values of QBA are the biggest and the values of QLSD equal zero. These results indicate that MDSS has perfect adaptability. As far as MDSR is concerned, because it employs the Round-Robin method in step 2, so the QLSD of MDSR basically keeps invariable. Fig.4(e) shows that the values of the makespan of MDSS and MDSR decrease when the node number increases. That is because MDSS and MDSR exploit the load balancing method, which makes the system have higher throughput. Fig.4(f) illustrates that with the increase of node number, MDSS, MDSR and DASAP have better load balancing except DALAP, The reason is that DALAP allocates a task to the node on which it has the latest start time, making the system unbalanced. 5.3 Performance Impact of Power Span In this set of experiments, we examine how the power span that decides the node heterogeneity affects the performance of MDSS, MDSR, DASAP and DALAP. The experimental results are illustrated in Fig.5.

10 Xiao-Min Zhu et al.: Multi-Dimensional Scheduling for Real-Time Tasks on Heterogeneous Clusters 443 Fig.5. Performance impact of power span. Fig.5(a) plots that MDSS and MDSR always have higher GRs than DASAP and DALAP, which can be explained as shown in previous subsection that MDSS and MDSR both set the QoS levels of tasks minimal in the first step, but DASAP and DALAP arbitrarily select QoS levels. Therefore, MDSS and MDSR obtain higher schedulability. From Figs. 5(b) 5(d), we can observe that QBA, QLA and QLSD of MDSS improve with the increase of node heterogeneity because the accepted task number slightly decreases with the increase of node heterogeneity. Although the node heterogeneity is changed, the whole processing power of nodes is changeless in our experiments, so the accepted tasks have more slack time to be used by MDSS to enhance the QoS levels. Another observation from Fig.5(d) is that MDSR has minimum QLSA, which seems that MDSR is able to get better QoS fairness. But, Figs. 5(b) 5(c) clearly show that the QBA and QLA of MDSR is the lowest. The reason is that adopting Round-Robin method makes a long-lived task enhance one QoS level resulting in many short-lived tasks that cannot improve their QoS levels. However, MDSS strives to maximize the QoS levels for the whole tasks by our QoS enhancement mechanism. Fig.5(e) shows that the makespan of MDSS slightly decreases with the interpretation that the accepted tasks decrease. However, we can observe that the makespan of DALAP still keeps invariable despite less accepted tasks, which reveals that DALAP has poor flexibility. Fig.5(f) plots MDSS, MDSR and DASAP can guarantee the system has better load balancing regardless of the change of node heterogeneity, but DALAP shows the worse load balancing with the increase of node heterogeneity. All the experiments in this group reflect that the node heterogeneity has slightly affected MDSS. Though the GR decreases a little, the QBA, QLA, QLSD and makespan are improved reciprocally, and the load is still balanceable. As a result, MDSS has perfect adaptability. 5.4 Performance Impact of Base Deadline This subsection focuses on the performance impact of base deadline that determines tasks have loose deadlines or not. We want to see when the tasks have loose time, how the strategies and algorithms use the slack time. Fig.6(a) shows that when the deadlines become loose, MDSS, MDSR, DASAP and DALAP all boost up GRs. This is because the time constraints are not tight as before. At the same time, MDSS and MDSR have higher GRs than DASAP and DALAP, which is as explained in the first group of experiment. Even though tasks have more slack time when their deadlines become loose, we can observe from Figs. 6(b) 6(d) that DASAP and DALAP do not improve QoS levels of

11 444 J. Comput. Sci. & Technol., May 2009, Vol.24, No.3 Fig.6. Performance impact of base deadline. tasks, because they do not adopt any QoS enhancement mechanism, so the slack time of accepted tasks are wasted. However, MDSS makes full use of the slack time to enhance QoS benefit with the time constraints, thus we can observe from Figs. 6(b) 6(d) that the QBA and QLA increase and the QLSD decreases with the increase of base deadline. Fig.6(e) shows that the makespans of MDSS, MDSR, DASAP and DALAP increase when the value of base deadline increases from 20 to 50 because of more accepted tasks. However, the makespan of MDSS basically keeps a constant value and noticeably outperforms those of others policies when the base deadline is more than 50. That is because we use the ITLB algorithm in step 3 of MDSS, which makes the system have higher throughput. Fig.6(f) reveals that with the increase of base deadline, MDSS, MDSR and DASAP all have better load balancing, but the DALAP allocates more tasks to some nodes and other nodes have fewer tasks, which makes the system unbalanced. 5.5 Performance Impact of Base Time This experiment is intended to investigate the performance impact of base time that reflects the granularity of tasks. From Fig.7(a), we can observe that when the base time increases, the GRs of all policies decrease. This is because tasks have longer execution time for bigger task heterogeneity. If the execution time becomes longer, those tasks whose execution orders are later will have later earliest start time, but the deadlines of them are not changed. Thus, some more tasks are rejected because of not satisfying their timing constraints. Even though all the GRs decrease, MDSS and MDSR always have higher GRs than DASAP and DALAP by average of 18.56% and 15.07%, respectively in our experiments. Figs. 7(b) 7(d) show that the QBA, QLA and QLSD of MDSS and MDSR become worse with the increase of base time. The main reason behind this is that the time of enhancing QoS levels increases for the bigger granularity of tasks, thus, only seldom tasks can have higher QoS levels within timing constrains. Because MDSR uses the Round-Robin method and does not consider the difference among tasks. So we can observe that MDSS has better performance than MDSR in these metrics. Fig.7(e) shows that when the base time increases, the makespans of all policies increase because of the longer execution time with the increase of task granularity. Even so, we also find that MDSS is better than others. Fig.7(f) illustrates that MDSS, MDSR and DASAP have better load balancing regardless of the granularity of tasks.

12 Xiao-Min Zhu et al.: Multi-Dimensional Scheduling for Real-Time Tasks on Heterogeneous Clusters 445 Fig.7. Performance impact of base time. 6 Conclusions In this paper, we have proposed a scheduler model which is efficient to provide multiple performance guarantees for QoS-aware real-time applications running on heterogeneous clusters. According to the scheduler model, a novel scheduling scheme called MDSS is investigated in this paper. The MDSS scheme adopts three steps. In Step 1, any existing real-time scheduling algorithm can be used to satisfy the timing constraint, which greatly improves the flexibility and expansibility of the system. In this paper, we use DASAP algorithm in Step 1. Two novel heuristic algorithms called MQFQ and ITLB are proposed and used in Steps 2 and 3, respectively. The MQFQ is a fair algorithm that makes all the tasks in one node have higher QoS levels, and the difference of QoS levels is minimal on the basis of Step 1. In addition, this QoS level enhancement method will not produce local extremum. The ITLB algorithm makes the system achieve load balancing by minimizing the standard deviation of nodes finish time. We compare the MDSS scheme with DASAP, DALAP algorithms and MDSR scheme, and the experimental results show that MDSS outperforms others with respect to guarantee ratio, QoS benefit, throughput and load balancing. Future studies in this area are three folds. First, to make the scheduling results more precise, we will combine the communication and dispatching times into the MDSS strategy. Second, based on the MDSS strategy, a fault-tolerant scheduling scheme will be investigated, and primary/backup versions will be our approach. Third, we plan to study a more complex version of MDSS, in which reliability issue will be taken into account. Acknowledgement We are grateful to the anonymous referees for their insightful suggestions and comments. References [1] Hwang K, Xu Z. Scalable Parallel Computing: Technology, Architecture, Programming. USA: McGraw-Hill, [2] Qin X, Jiang H. A dynamic and reliability-driven scheduling algorithm for parallel real-time jobs executing on heterogeneous clusters. Journal of Parallel and Distributed Computing, 2005, 65(8): [3] Xie T, Qin X. Scheduling security-critical real-time applications on clusters. IEEE Trans. Computers, 2006, 55(7): [4] Krishna C M, Shin K G. Real-Time Systems. USA: McGraw- Hill, [5] Zhu X, Lu P. Study of scheduling for processing real-time communication signals on heterogeneous clusters. In Proc. 9th Int. Symp. Parallel Architectures, Algorithms, and Networks, Sydney, Australia, May 7 9, 2008, pp [6] Zhu X, Lu P. Scheduling of real-time signal processing in cluster-based software radio systems. Journal of Software, 2009, 20(3): [7] Zhu X, Lu P. Multi-dimensional scheduling scheme for QoS-

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IEEE Trans. Parallel and Distributed Systems, 2006, 17(11): [29] Harada F, Ushio T, Nakamoto Y. Adaptive resource allocation control for fair QoS management. IEEE Trans. Computers, 2007, 56(3): [30] He L, Jarvis S A, Spooner D P. Dynamic scheduling of parallel jobs with QoS demands in multiclusters and grids. In Proc. 5th IEEE/ACM Int. Workshop on Grid Computing, Pittsburgh, USA, Nov. 8, 2004, pp [31] Doǧan A, Özgüner F. On QoS-based scheduling of a meta-task with multiple QoS demands in heterogeneous computing. In Proc. 16th IEEE Int. Symposium on Parallel and Distributed Processing, Florida, USA. Apr , 2002, pp Xiao-Min Zhu received the B.S. and M.S. degrees in computer science from Liaoning Technical University in 2001 and 2004, respectively. He is currently working toward the Ph.D. degree with the School of Computer Science, Fudan University, China. His research interests are high performance computing, cluster computing, fault-tolerant computing, and information security. Pei-Zhong Lu received the B.S. and M.S. degrees in applied mathematics from Institute of Information Engineering in 1982 and 1987, respectively. He received the Ph.D. degree from Institute of Systems Science, AMSS, China, in He is currently a professor in the School of Computer Science at Fudan University. His research interests are information security, algebra coding, parallel and distributed systems, real-time computing and performance evaluation. He has published more than 50 technical papers in reputable journals and conference proceedings. He received the Excellent Doctoral Dissertation Award of China in He is a member of IEEE.